Specifically, as an intermediate step, Proposition REF  in Appendix  shows that the difference between the c.d.f. of (REF ), scaled by \(s_{tk} = \sqrt{\Omega _{t,k}^{*2} + \Omega ^*_{t,t}\Omega ^*_{k,k}}\) , and that of a standard Gaussian random variable is bounded by \(\frac{s_1\log (K\vee n)}{n^{1/2}}\) . Therefore, asymptotic normality holds as soon as this quantity converges to zero, which agrees with the weakest sparsity conditions for Gaussian graphical model inference in the literature [1]}, [2]}. In addition, the asymptotic variance \(s_{tk}^2\)  agrees with the minimum variance bound in Gaussian graphical models [2]}. Thus, inference based on the empirical risk function (REF ) does not lead to any asymptotic efficiency loss. Unlike the previous works, we do not require the bounded operator norm condition, \(\lambda _{\max }(^*)\le C\) . This condition is avoided in our analysis by using a more convenient empirical risk function (REF ), as opposed to the log-likelihood in [4]}, and a CLIME-type initial estimator (REF ) satisfying (REF ), as opposed to the node-wise Lasso estimator in [2]}.
